Game theory to enhance stock management of personal protective equipment (PPE) during the COVID-19 outbreak
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Game theory to enhance stock management of personal
protective equipment (PPE) during the COVID-19 outbreak
Khaled Abedrabboh1 , Matthias Pilz2 , Zaid Al-Fagih3 , Othman S. Al-Fagih4 ,
Jean-Christophe Nebel5 , Luluwah Al-Fagih1,5*
1 College of Science and Engineering, Hamad Bin Khalifa University, Doha 34110, Qatar
2 Independent Researcher, e-mail: Pilz.Matthias@outlook.com
3 Independent Researcher, e-mail: zaid.al-fagih@nhs.net
arXiv:2009.11838v2 [cs.CY] 25 Sep 2020
4 NHS Health Education East of England, Cambridge, CB21 5XB, UK
5 School of Computer Science and Mathematics, Kingston University, London KT1
2EE, UK
*Corresponding author. Email: lalfagih@hbku.edu.qa
Abstract
Since the outbreak of the COVID-19 pandemic, many healthcare facilities have suffered
from shortages in medical resources, particularly in Personal Protective Equipment
(PPE). In this paper, we propose a game-theoretic approach to schedule PPE orders
among healthcare facilities. In this PPE game, each independent healthcare facility
optimises its own storage utilisation in order to keep its PPE cost at a minimum.
Such a model can reduce peak demand considerably when applied to a variable PPE
consumption profile. Experiments conducted for NHS England regions using actual data
confirm that the challenge of securing PPE supply during disasters such as COVID-19
can be eased if proper stock management procedures are adopted. These procedures can
include early stockpiling, increasing storage capacities and implementing measures that
can prolong the time period between successive infection waves, such as social distancing
measures. Simulation results suggest that the provision of PPE dedicated storage space
can be a viable solution to avoid straining PPE supply chains in case a second wave of
COVID-19 infections occurs.
1 Introduction
Novel infectious diseases pose a serious challenge to policy makers and healthcare systems.
Emerging from Wuhan, China, the ongoing Coronavirus Disease 2019 (COVID-19)
pandemic, caused by the Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-
2), has wreaked havoc globally. This is due to its rapid rate of transmission, its virulence
and the inability of most countries to adequately prepare for such a disease [1]. Identified
in December 2019, the disease now has a global distribution with over 30 million confirmed
cases and almost one million confirmed deaths as of September 2020 according to the
World Health Organisation (WHO) [2]. COVID-19 is primarily transmitted through
respiratory droplets and the WHO has identified two principal routes through which
these are carried between people. The first mode of transmission involves a person being
in direct, close contact with someone who carries the virus (within one metre) where they
become directly exposed to potentially infectious respiratory droplets. The second mode
of transmission involves contact with fomites in the immediate vicinity of the infected
September 28, 2020 1/21
person [3]. Nguyen et al. estimated that frontline healthcare professionals (HCPs) had
a 3.4 times higher risk than non-healthcare workers of contracting COVID-19, even
when adjusting for the probability of being tested [4]. Indeed, approximately 10% of
the confirmed cases in China [5] and up to 9% of all cases in Italy have been among
healthcare professionals [6] as of the date of publication of these studies. This increased
risk does not only pose a problem for the HCPs themselves, but also poses a major
threat to the elderly and vulnerable populations they care for, since outbreaks within
healthcare settings are important amplifiers of infection [7]. One of the most crucial ways
by which transmission of this virus (and other infectious diseases) is reduced is the use
of proper Personal Protective Equipment (PPE) [7]: the WHO recommends a surgical
mask, goggles, or face shield, gown, and gloves to be worn as PPE in their COVID-19
PPE guidelines. If an aerosol-generating procedure is performed, the surgical mask is
replaced with an N95 or FFP2/3 respirator which provides a greater level of filtration
than surgical masks [8]. These guidelines are replicated globally with minor differences.
In the United Kingdom (the case study reported in this paper), the most recent Public
Health England guidelines are essentially the same [9]. Globally, medical resources,
particularly PPE have come under unprecedented demand. This has led to shortages
in many countries with some rationing their use of PPE and in some cases reusing
disposable material [10]. The natural consequence of this has been the disproportionately
high rate of infection amongst HCPs which in turn contributes to disease spread. This
may not necessarily result from a national shortage of PPE but rather local shortages
resulting from inefficient distribution of resources in timely manner [11].
In spite of the limited applicability of resource allocation methods, as choices have to
be made, they can aid in making the decisions that achieve the best health outcomes
(see [12] for a review on the use of such methods in epidemic control). Single- and
multi-objective optimisation methods have been used frequently to address problems of
resource allocation and scheduling of purchase orders for medical supplies (see [13] for a
comprehensive survey and taxonomy). Such methods are centralised in the sense that
a central planner seeks to optimally coordinate supply activities for the entire system,
e.g. minimise overall costs for the central planner. In contrast to this, game theory is
a tool that allows decentralised decision-making. That means, different entities in the
system can make decisions based on their individual preferences. For instance, they can
schedule their orders to minimise their own costs. The decentralised approach leaves the
actors more freedom and is the more applicable direction for our scenario.
Game theory has been applied to a variety of subject areas such as biology [14],
economics [15], computer science [16] and energy [17, 18]. In general, game theory is
used to mathematically model systems of competing agents. Usually these individuals
act in a selfish and rational manner. In this context, selfish can be understood as
being only interested in their own good, i.e. an agent strives to maximise its outcome
irrespective of the outcome of others. Furthermore, rational means that there is a clear
logical reasoning behind every decision. The most widely used solution concept for a
non-cooperative game is the Nash equilibrium [19]. It is achieved when none of the
players has an incentive to change their strategy unilaterally.
Despite its various applications in supply chain management (cf. [20] and the references
therein), only few studies have applied game theory to the management of medical
supplies. To the best of our knowledge, a 2008 study [21] (an extended version was later
published in [22]) was the first to develop a game theoretic approach for stockpiling of
critical medical items. In their model, the authors propose a non-cooperative game where
hospitals stockpile critical medical resources in preparation for disasters. It is proposed
that these hospitals determine their individual stockpile levels strategically in a way that
minimises their expected total spend. Their model uses a cost function that consists
of the cost of ordering, borrowing, storage and a penalty cost in the case of shortages.
September 28, 2020 2/21
Although their model proves to give some breathing space to the medical supply chain
at the onset of an epidemic, it is only intended as a preparation scheme, and therefore
may suffer from inapplicability if an epidemic lasts longer than the planning period
considered. The model assumes a given likelihood of the occurrence and severity of a
pandemic, which is, in practice, heavily unpredictable. Their work was further developed
in [23] by introducing network constraints, thus giving realistic hospital sharing policies.
The authors find that deficits between stockpiles and demand can be reduced through
central stockpiling and through increasing penalties for deficits.
Game theory has also been applied to the problem of drug allocation [24], where
countries behave selfishly to minimise their expected number of infections. Unlike in [25]
where the authors propose a similar resource competition among countries, their model
imitates the stochasticity of infection transmission parameters. The resulting comparison
between this selfish allocation scheme and a utilitarian division scheme, where a central
planner (e.g. WHO) makes all of the allocation decisions, shows that having a central
planner reduces the total number of worldwide infections considerably. Although their
proposed model can aid in understanding how countries, acting in their own self interest,
would behave in an epidemic, it does not address the problem of resource distribution
within a given country. [26] compares selfish vaccination coverage, which the authors
call the ‘Nash vaccination‘, with group optimal coverage, which they call the ‘utilitarian
vaccination’. The authors find that the cost of vaccination would be a pivotal factor in
determining the effectiveness of Nash vs. utilitarian coverage. However, this can result
in different outcomes depending on the level of disease severity according to age, such as
in the case of chickenpox.
Most recently, Nagurney et al. proposed a novel decentralised model for medical
supply chain with multiple supply and demand points [27]. In their model, selfish
consumers who have stochastic demands make purchasing decisions that minimise their
total expenditure. The authors assume that the disutility function of a consumer
consists of a linear cost of demand, a quadratic cost of transportation and a penalty for
shortages/surpluses. Although the quantification of the shortage/surplus penalty can be
debated, the authors conclude by suggesting that shortages in supply can be avoided by
redirecting global production efforts to instead adding or increasing local production
capabilities.
In this paper, we propose a game theoretic approach for managing PPE supplies
during a pandemic. We take inspiration from the electricity storage scheduling game
developed by the co-authors in [28], where a decentralised system of individually owned
home energy systems served by the same utlity company schedule their day-ahead battery
usage over a full year. This decentralised system resulted in improved energy efficiency
for the utility company and cost savings for the participating users.
This work proposes a centralised-decentralised approach to the PPE supply chain
(cf. Section 2). In the proposed architecture, healthcare facilities report their demand to a
central entity. This central entity is assumed to have the commitment power to fulfill its
orders and set the costs for PPE. Given the actions of all game participants, healthcare
facilities optimise their PPE orders by making stockpiling decisions individually. Our
approach is centralised in the sense that cost and supply is controlled by a central entity.
It is also decentralised because healthcare facilities make their stockpiling decisions
independently. By adapting the model developed in [28] and applying it to COVID-19
related PPE demand in England, we study the effects of early stockpiling as well as
increasing storage capacities on PPE supply in challenging circumstances. Additionally,
we examine the impact of a putative second wave on PPE supply and investigate whether
delaying this putative second wave can ease the challenge of fulfilling the PPE demand
of healthcare facilities. Finally, insights and conclusions from this study are drawn and
suggestions regarding PPE supply in disaster management are discussed.
September 28, 2020 3/21
The contributions of this work can be summarised as follows:
1. A game theory-based model designed to enhance stock management of PPE
supply.
2. A study of PPE supply management in England during the current COVID-19
pandemic demonstrating the benefits of the centralised-decentralised allocation
approach advocated by the proposed model.
3. A detailed analysis of how key factors, i.e. stockpiling start date, storage capacity
and the date of a putative second wave, impact PPE supply in England.
4. Insights and suggestions regarding the handling of a putative second wave in
terms of PPE supply management.
The remainder of this paper is organised as follows. In Section 2 we provide an
overview of the system architecture, explain the chosen cost function and give details on
the game formulation. Section 3 contains information on the experimental setup and
presents an analysis of the results with simulations of different scenarios. This is followed
by a discussion and recommendations in Section 4 and finally, Section 5 concludes the
paper and points out future research directions.
2 Materials and methods
During the COVID-19 pandemic, most countries have experienced strained healthcare
resources and even shortages. The efficient management of personal protective equipment
(PPE) supply has proved vital in limiting the spread of the virus and in keeping the
healthcare professionals protected. In this paper, after designing a game theory-based
model to enhance stock management of PPE supply, we investigate as a case study
PPE provision in hospitals of the English National Health Service (NHS). England was
selected as not only has it been one of the first and most severely hit countries, but it
has also released COVID-19 data in a transparent and timely manner.
System architecture
Healthcare facilities in England are run by the NHS (NHS England). The NHS is made
up of organisational units named Trusts which mainly serve geographic areas, but can
also serve specialised functions. For a detailed description of the NHS structure in
England, please see [29]. Prior to the outbreak of COVID-19 in England, procurement
of medical supplies to NHS Trusts, including PPE, was centralised where each Trust
would ‘order’ supplies via NHS Supply Chain [30]. However, in the initial period of the
pandemic, the NHS was unable to fulfil Trusts’ demand centrally leading to a chaotic
albeit temporary decentralised supply chain; some of which was unconventional [31]. On
01 May, NHS Supply Chain introduced a dedicated PPE supplies channel separate to
other medical supplies to address this issue [32].
The NHS openly expresses its commitment to improve efficiency [33] and, via its
NHS Improvement department, it has advocated the use of modelling and novel ideas to
facilitate this [34]. In line with favoured practice by the NHS, we propose a centralised-
decentralised PPE supply chain architecture, shown in Fig 1, where a central entity
(controlled by the NHS) has the commitment power to set the national cost of PPE
based on the market price of the sourced PPE and to make PPE deliveries as ordered
by the NHS Trusts. In this architecture, NHS Trusts, each selfishly concerned with their
own interests of satisfying their demand and minimising their cost, manage the use of
September 28, 2020 4/21
their storage capabilities so that their PPE cost is minimised. After optimising their
storage schedules independently, NHS Trusts then report their optimal PPE orders to
the central entity, which in turn fulfils their demand and charges them their share of
PPE cost.
Fig 1. System architecture for the proposed model, showing the
centralised-decentralised approach to PPE supply chain.
Cost function
In order to ensure security of supply and avoid burdening the supply chain, sudden
surges in demand should be eliminated or at least planned for. It is therefore beneficial
for the central entity to aim at having a relatively flat demand profile. This can be done
by incentivising Trusts to utilise their storage in an optimal manner so that their overall
demand is level. Accordingly, the following assumptions are made regarding the cost
function of PPE, C(Q), where Q is the total quantity of PPE consumption:
1. The cost function of PPE C(Q) is strictly increasing in consumption Q,
dC
> 0, (1)
dQ
i.e. the higher the consumption the higher the cost.
2. The marginal cost of PPE is strictly increasing in consumption,
d2 C
> 0, (2)
dQ2
i.e. the rate of rise of cost increases when consumption increases. This assump-
tion can be justified by considering that an increase in demand will likely require
new supply routes or new production facilities be established.
September 28, 2020 5/213. Zero consumption yields (incurs) zero cost
C (0) = 0. (3)
While many functions satisfy the above assumptions, this work adopts a quadratic
cost function where
C(Q) = aQ2 + bQ (4)
and where a > 0, b ≥ 0 are constant cost parameters, as is the case in [27], where the
authors use a quadratic cost function for transportation of PPE along with a linear cost
function for PPE supply. The use of quadratic cost function of PPE is further supported
by the documented rise in PPE costs during peak COVID-19 cases in England. In some
instances, a 1000% increase in prices was reported by NHS Trusts [35].
Game formulation
Within our model the consumption Q (as introduced in the previous section) consists
of two separate quantities. On the one hand, there is the demand d for PPE according
to the number of patients that are currently treated. This number cannot directly be
influenced within the game formulation. On the other hand, there is the number of PPE
kits a that are put into the storage (or taken from the storage), which is our decision to
make. Thus we have:
Q=d+a . (5)
While d will always be larger than zero, a can take values in a range from max(−d, −s)
to smax − s. Formally, this can be summarised by the following constraint:
a − (smax − s)
h(s, a) = . (6)
−a + min(d, s)
That means, the largest amount of PPE that can be taken from the storage is limited by
the current demand and the amount of available stored PPE. The upper bound is due
to the finite amount of space in the storage facilities. The chosen action a then directly
affects the stored PPE leading to the following transition equation:
f (sold , a) = snew = sold + a . (7)
Similar to [28], we propose a discrete time dynamic game, where the decisions of the
players (the Trusts) of how much to put/take in/from the PPE storage are performed
sequentially in stages. These stages (also called intervals) are defined according to the
actual demand variations, i.e. if the demand changes on a daily basis, each interval
would cover a one day period. Furthermore we introduce the state of the game (for each
interval) and how it interacts with the decisions of the players. Overall the goal of the
players is to minimise their own costs of PPE, i.e. their utility function which is closely
related to the cost function discussed in the previous section (see Eq 4). To summarise,
the game consists of:
1. A set of players (Trusts) N = {1, . . . , N }, where N is the total number of
participants. All players are assumed to be selfish and rational.
2. A set of intervals (days), T = {0, . . . , T − 1} where T is the number of intervals
that comprise the intended time cycle of the game.
September 28, 2020 6/213. Scalar state variables stn ∈ Sn ⊂ IR denoting the amount of stored PPE of the
nth player at stage t ∈ T ∪ {T }. Collectively, we denote the state variables of
all players at stage t by st := [st1 , . . . , stN ] ∈ S := S1 × · · · × SN ⊂ IRN . In the
open-loop information structure it is assumed that the initial state s0 is known
to all players n ∈ N .
4. Scalar decision variables atn ∈ Hnt (stn ) ⊂ An ⊂ IR (for definition of Hnt see
item (5)) denoting the usage of the stored PPE of the nth player at time
t ∈ T . Collectively, we denote the decision variables of all players at stage t
by at := [at1 , . . . , atN ] ∈ A := A1 × · · · × AN ⊂ IRN . Furthermore we define the
schedule of PPE usage of an individual player n ∈ N as a collection of all its
decisions in the stages of the game by an := a0n , . . . , aTn −1 . A strategy profile
is denoted by a := [a1 , . . . , aN ].
A set of admissible decisions Hn s0n := {an | htn (stn , atn ) ≤ 0, t ∈ T } ⊂ IRT
5.
for the nth player. The function htn (stn , atn ) has been defined in Eq 6, cap-
turing the restrictions posed by the storage facilities. We denote Hnt (stn ) :=
{atn | htn (stn , atn ) ≤ 0} ⊂ IR
6. A state transition equation
st+1 = fnt stn , atn , t ∈ T , n ∈ N ,
n (8)
T
governing the state variables {st }t=0 . The function fnt (stn , atn ) is the discretised
version of the transition equation (Eq 7), showing how a decision of the player
influences the state of its PPE storage for the upcoming stage.
7. A stage additive utility function
−1
TX
un s0n , [an , a−n ] = −CnT sTn − Cnt stn , atn , at−n
(9)
t=0
for the nth player, where a−n := [a1 , . . . , an−1, an+1 , .
. . , aN ] denotes the deci-
sions of all other players. The function Cnt stn , atn , at−n fulfils the assumptions
as denoted in Eq 4 capturing the costs to the nth player at the tth stage. Note
that the utility function depends only on the initial state variable s0n , since the
subsequent states stn are determined by Eq 8. The function
CnT sTn = sTn
(10)
can be interpreted as a penalty for the nth Trust that is incurred by ending
up in state sTn , i.e. its overbought PPE capacity, at the end of the scheduling
period.
The objective of rational players is to maximise their total utility, i.e. minimise
their overall costs, over the complete time cycle T . We represent the decision problem
Gn of the nth player (given the actions a−n of all the other players) as the following
optimisation problem:
Gn (a−n ) given s0 ∈ S
maximise un s0n , [an , a−n ]
an
(11)
subject to atn ∈ Hnt stn
st+1 = fnt stn , atn
n ∀t ∈ T ∪ {T }
Moreover, the game is referred to as {G1 , . . . , GN }, which denotes the simultaneous
September 28, 2020 7/21(and linked) decision problems for all the Trusts. Within this game formulation we can
formally define the Nash equilibrium (cf. Section 1) by:
A strategy profile â = [â1 , . . . , âN ] is a Nash equilibrium for the game
{G1 , . . . , GN } if and only if for all players n ∈ N we have
un s0n , [ân , â−n ] ≥ un s0n , [an , â−n ] , ∀an ∈ Hn s0n .
(12)
3 Results
Data sets and experimental setup
In this paper, we propose a game-theoretic approach for the supply of PPE to healthcare
facilities. We consider the case study of England as it was one of the countries severely
impacted by the COVID-19 pandemic and where shortages in medical resources, especially
PPE, were reported [10]. The players in this game structure are assumed to report
their demand and are committed to paying their respective invoices. The players are
also assumed to have the financial independence that inspires their selfish behaviour.
Although this game is intended for independent healthcare providers, such as NHS Trusts
in England, only region-level COVID-19 data are publicly available. Therefore, in this
experiment, we assume that each of the seven NHS England regions can act selfishly in
a way that represents the interests of its accommodated Trusts. We believe this has no
effect on the insights drawn from this experiment since a region is merely a group of
Trusts.
Demand profiles until 01 Aug
The region demand profiles used in this experiment were originated from daily COVID-
19 occupied hospital beds data, which is available for the seven NHS England regions
and published by the UK government [36]. Fig 2 shows the peak and total (up to 01
Aug) COVID-19 occupied hospital beds for each of the seven NHS England regions.
We assume this information can represent the extent of the regions’ response to the
pandemic. Therefore, we assume that the PPE consumption that is related to all
COVID-19 activities within these regions can be estimated from their daily COVID-19
occupied hospital beds data.
On 15 Apr, the UK government published its estimated PPE deliveries in England
since the start of the outbreak in the country: “Since 25 February 2020, at least 654
million items of PPE have been supplied in this way” [37]. As this number counts a
pair of hand gloves as two items and includes consumables that are not considered for
this study as mentioned in Section 1, e.g. body bags and swabs, we used the number of
delivered aprons (135 M) as our reference for consumed PPE kits. However, only 86%
of these items were delivered to the NHS Trusts, whereas the rest were distributed to
primary care providers (GPs), pharmacies, dentists and social care providers, as well
as to other sectors [37]. Additionally, although these deliveries took place during the
period between 25 Feb to 15 Apr, the first COVID-19 hospitalization only took place on
20 Mar. Also, given that PPE supply chains were heavily burdened at the beginning of
the outbreak, the UK government was led to change PPE usage guidance that health
and social care workers should use in different settings when caring for people with
COVID-19. The most notable of these was on 02 Apr when a single kit of PPE was
advised to be used per session rather than per patient [38]. This change of guidance
along with the fact that our model is only concerned with COVID-19 related PPE
consumption have led us to assume that 50% of the PPE kits delivered to NHS Trusts
between 25 Feb to 15 Apr were consumed for COVID-19 related activities in the period
September 28, 2020 8/21Fig 2. Total and peak occupied hospital beds with illness related to COVID-19 in the
seven regions of NHS England up to and including 01 Aug 2020.
between 20 Mar and 15 Apr. This results in an estimated PPE consumption of 58 M
kits for COVID-19 related hospitalisation cases. Dividing this number by the aggregated
daily COVID-19 occupied beds in England for the aforementioned period results in a
rough estimate of 195 PPE kits daily consumption per occupied hospital bed. Taking
this into consideration, along with the mentioned change in guidance, we were able to
extrapolate the publicly available daily COVID-19 occupied beds region data into PPE
consumption by using a factor randomised between 210 and 240 kits per bed until 02
Apr, and between 150 and 180 from then onwards.
Storage Capacity
According to the experience of our physician co-authors who have worked in several NHS
hospitals, it can be estimated that each set of 20 hospital beds is supported by a PPE
storage space of three shelves, the dimensions of each are 4.0 m × 0.6 m × 0.8 m. Our
extrapolation in m3 to the storage capacity of the seven NHS England regions is listed in
Table 4. Moreover, using the volumes supplied by PPE suppliers for bulk orders [39–42],
the volume of a thousand of PPE kits, each consisting of a face shield (visor), a face
mask, an apron and a pair of gloves, can be approximated to 1.35 m3 . Consequently,
the storage capacities in terms of PPE kits can be estimated for each region as seen in
Table 4.
Although we are aware that our estimates are very coarse, we believe that they
represent a reasonable attempt at quantifying those largely unpublished quantities. In
September 28, 2020 9/21any case, they are only used as a baseline for our modelling as four other values of
storage capacities are also considered for each region, i.e. the baseline storage multiplied
by a factor of 5, 10, 15 and 20.
Table 4. Storage capacities of the seven NHS England regions.
Region Maximum COVID-19 Storage Storage in
occupied beds in m3 PPE kits
East of England 1, 484 427 316, 590
London 4, 813 1, 386 1, 026, 770
Midlands 3, 101 893 661, 550
North East & Yorkshire 2, 567 739 547, 630
North West 2, 890 832 616, 530
South East 2, 073 597 442, 240
South West 840 242 179, 200
Second wave
Given that several countries have already experienced a second wave of COVID-19
infections (e.g. Spain and France) [43] and several studies [44–46] predict that this is
most likely to occur in England as well, our model is used to investigate how PPE
demand should be handled in such a situation. Those simulations rely on the model
proposed by [44], where the authors predict that a second surge of COVID-19 cases
happening in England can be of a lesser magnitude than the first wave, depending on
the control measures in place (e.g. lockdown and social distancing measures), but will
most likely last longer than the first wave.
Therefore, a second surge in PPE demand was introduced to the regions’ demand
profiles from 01 Aug. Fig 3 shows an examples of a second wave PPE demand profile
expected to peak in mid October 2020. As the time of occurrence of a putative second
wave is largely unknown, we considered in our experiments waves peaking at five different
dates, i.e. mid October, mid November, mid December 2020 and mid January and mid
February 2021. As can be shown from [44], it is highly unlikely that a putative second
wave would last longer than 100 days after peaking. This has led us to assume that PPE
demand for the second wave would come to an end 100 days after the second peak.
Stockpiling starting dates
In this study, we investigate whether shortages in PPE could have been avoided if the
stockpiling game had been initiated at an earlier stage. Five different starting dates
are considered as each of these had some importance regarding the development of the
COVID-19 pandemic in the UK. Consequently, each could have triggered the start of
the PPE stockpiling process:
1. 20 March 2020: data on COVID-19 cases and occupied hospital beds in England
start to be released to the public [36].
2. 11 March 2020: the WHO declared COVID-19 a pandemic [47].
3. 28 February 2020: the European Union proposed to the UK a scheme to
bulk-buy PPE [48].
4. 07 February 2020: WHO warned of PPE shortages [49].
5. 31 January 2020: the first COVID-19 case was confirmed in the UK [50].
September 28, 2020 10/21Cost parameters
As mentioned in Section 2, considerable rises in PPE costs were reported during peak
COVID-19 cases in England [35]. This led us to use a quadratic cost function in our
model to express cost as a function of demand (see Eq 4). Since our proposed cost
function does not have any cost terms other than the cost of ordering PPE, scheduling
decisions are fairly independent of the cost parameters. In fact, the authors in [28]
showed that the sensitivity of similar scheduling games in relation to cost parameters is
quite low. In the experiments conducted in this study, we use 8 × 10−6 as the quadratic
term parameter and 1 × 10−2 as the linear term parameter. Usage of these parameters
has resulted in a 300% increase in PPE cost between peak and average demand. We
believe this is in line with what has been reported in the literature [35], where the costs
of certain PPE items increased by up to 1000% between pre-COVID and peak COVID
periods.
Experiment implementation
In order to show the outcome of the game and whether it has the capability to ease the
challenge of securing PPE during peak demand, this experiment was implemented in two
stages. In the first stage, we considered the overall period using five different stockpiling
start dates (as stated above), to the end of a putative second wave, which also has five
different peak dates. Additionally, we used five different storage capacities for each
region in order to analyse the effects of adding extra PPE storage space. This means
that 125 games were played in the first stage, the results of which will be discussed in the
following sections. In the second stage of this experiment, we only examined the effects
of running the game for the duration of the second wave; with the simulation starting
from 01 Aug, (i.e. the cut-off date of the data used to simulate demand), and ending 100
days after the peak of the second wave. We have simulated five different dates for the
peak of the second wave. Furthermore, we used the same five storage capacities for this
stage as already employed before. The results of the 25 games that were simulated in
the second stage will be discussed in the following sections. In both stages, we assume
that regions start the scheduling game with an empty store.
A Python code was written to find the Nash equilibrium of this game, where players
sequentially update their PPE orders to minimise their costs. The optimisation package
scipy.optimize was used for each player to find their optimal scheduling decisions at
each game iteration. The game converges once the mean squared error (MSE) between
successive game iterations drop below a threshold. Given that peak PPE demand is in
the order of millions of sets, we used a maximum value of 10 for the MSE to obtain
accurate results.
Analysis of outputs generated by the game
In order to illustrate the outputs generated by the game, we show results from running
one of the games in Fig 3. The outcome of the game is a series of daily decisions that
each English NHS region should take regarding their PPE requirements: order PPE, use
PPE from their storage or stockpile PPE in their storage. The result of those decisions
can be visualised by monitoring the status of their storage levels. Fig 3 displays the
amount of PPE sets available in the storage for each region (coloured areas), the PPE
demand (dotted line) and PPE orders (bold line). From the stockpiling date, a constant
set of PPE is ordered daily, leading to storage reaching full capacity at the start of the
first wave that triggers PPE demand. However, as storage capacity becomes rapidly
insufficient to meet the daily demand, available (stored) PPE decreases rapidly and
daily ordering increases until reaching a plateau around the period of the peak of the
September 28, 2020 11/21first wave. Eventually, storage becomes depleted and the daily order equals the daily
demand of a receding wave. Finally, when the daily demand is sufficiently low, storage is
repleted allowing English regions to face the putative second wave with full PPE storage
capacity. From then, patterns observed in terms of order and storage levels are similar
to that already described for the first wave.
Fig 3. Millions of PPE sets required daily nationally (dotted line), ordered daily
nationally (bold line) according to the game, and stored in each of the seven regions of
NHS England (coloured areas). Here, stockpiling started on 28 Feb, standard storage
capacity was multiplied by five and the peak of the second wave is expected to happen
in mid October. Stored PPE sets for each region is stacked on top of each other in order
to show the cumulative PPE stock of all regions to illustrate the game mechanism. Note
that the areas are scaled down by a factor of five in order to make the figure more
readable.
As this model assumes that the cost of ordering PPE is quadratic with respect to
the aggregated PPE order, the game aims at producing a level of orders as constant and
low as the conditions permit. Deviations from this ideal scenario generate additional
costs which corresponds in the proposed model to an increased challenge for the NHS at
delivering required PPE. This challenge is visible in Fig 4 where a comparison between
three different scenarios is shown. The first scenario can be considered as a reference
scenario where NHS regions report their PPE demand to NHS without scheduling their
storage usage. The second and third scenarios are outcomes from the game when regions
start stockpiling on 11 Mar and 07 Feb, respectively. One should also note that storage
capacity in the third scenario is multiplied by 10. In this figure, the cumulative cost
for the second and third scenarios is shown with reference to the first. As shown, a
9% saving resulted from the second scenario at the end of the scheduling period while
the third scenario resulted in a 38% saving. Since a higher cost saving means that less
fluctuations occur in demand, these figures can be directly used as a method to quantify
the challenge of securing PPE. In Fig 4, as in the remaining figures, colours have been
used to illustrate that level of challenge. The associated colour grading goes from dark
red to dark green, where dark red expresses a high challenge and dark green indicates a
relatively low challenge.
September 28, 2020 12/21Fig 4. Comparison between the outcome of three different scenarios, a reference
scenario (without scheduling), when stockpiling starts on 11 Mar and when it starts on
07 Feb and storage capacities are multiplied by 10. This shows that cumulative cost
saving can represent the challenge level of securing PPE supply. A colour grading is
applied to represent the level of this challenge, where dark red is used for the highest
challenge level and green for the lowest.
Scenario simulations
Using the proposed model, a range of different scenarios were considered to assess how the
challenge in terms of fulfilling PPE demand varies according to the amount of available
storage capacity, the stockpiling starting date and the peak date of a putative second
wave of COVID-19. In total, 125 scenarios were conducted by considering five different
values for each of the three parameters (see subsections above). Fig 5 summarises the
outcomes of these experiments. The figure reveals that in a two-wave pandemic, the two
most critical parameters are the stockpiling date and storage capacity. With the peak of
the first wave being estimated at 08 Apr [51], and the second wave taking place at least
six months after the first one, the timing of the second wave thereafter would have only
a moderate impact on easing the PPE challenge. On one hand, Fig 5 highlights that
if the storage capacity is low, an early stockpiling starting date hardly alleviates the
PPE challenge and has minimal, if not negligible, cost savings (leftmost column). On
the other hand, a high storage capacity only slightly mitigates a late stockpiling date
(bottom row). Indeed, only early stockpiling and a sizeable increase in storage (top right
area) would provide the right conditions to lower the PPE challenge overall and indeed
would result in considerable cost savings.
In order to focus on the future, an additional 25 scenarios were simulated focusing
on the PPE challenges associated with the available storage capacity and the peak date
of a putative second wave. Using the last date of bed occupancy data (01 Aug) as the
September 28, 2020 13/21Fig 5. Challenge in terms of fulfilling PPE demand delivery according to the amount
of available storage capacity (x-axis), the stockpiling starting date in 2020 (y-axis), and
the peak date of a putative second wave of COVID-19 (each cell in the grid is divided in
five stripes corresponding, from top to bottom, to peak dates in October, November,
December 2020, January and February 2021).
simulation starting date, five different values were considered for the two parameters
(see subsections above). The results of these 25 scenarios are shown in Fig 6. As shown,
while the storage capacity remains a key parameter, the impact of the date of the peak
of the second wave has become apparent. Indeed, as the period of study is shorter than
in the previous set of simulations and only a single wave of infection is considered, its
timing substantially affects the PPE challenge. Consequently, even if a large amount of
storage is available (two rightmost columns), only a late 2020 peak or later would be
handled well in terms of cost savings.
4 Discussion and recommendations
The model we have constructed to analyse the provision of PPE has important implica-
tions for the management of the pandemic hitherto. Firstly, we have shown objectively
that early stockpiling and increasing storage capacity would have helped to massively
reduce the costs associated with the containment of the pandemic. Secondly, the early,
sufficient and cheap provision of PPE would have had a substantial impact on the
ability to contain the pandemic, and protect both the most vulnerable patients and the
healthcare professionals that care for them [7].
Based on these findings and the mathematics underlying our approach, we have
identified the key elements that govern the aggregated demand (and therefore, the cost)
of PPE in our scenario. The two key parameters involved in saving costs by reducing the
September 28, 2020 14/21Fig 6. Challenge in terms of fulfilling PPE demand delivery according to the amount
of available storage capacity (x-axis) and the peak date of a putative second wave of
COVID-19 (y-axis).
demand for PPE in the context of a pandemic that were identified using this approach
are: (i) the storage capacity (for PPE) available to the health system. We have shown
that increasing the storage capacity enhances the ability to stockpile which in turn
helps flatten the demand curve (making it easier for suppliers to meet this demand in a
cost-effective way) ; and (ii) the date when stockpiling of PPE commences. We have
shown that earlier stockpiling would have saved costs and ensured adequate provision
of PPE for the first peak of the pandemic. These two factors will also prove crucial
when planning for a second wave. Of secondary importance is the ability to predict the
date of the second wave. This is because, providing the second wave occurs after the
new year, the degree of cost saving that occurs diminishes to negligible levels beyond
this point in time. Using our game-theoretic approach, we have found that the NHS
would be able to achieve all the cost-saving advantages if the second wave was to occur
in 2021. A practical implication of this is that policymakers may need to keep some
of the restricting measures that are in place to control the pandemic until beyond this
point in time.
In order to highlight the significance of having dedicated and sufficient storage for
PPE in preparation for a putative second wave, we show a comparison between different
storage capacities in Fig 7. In this figure, the aggregate demand profile for all NHS
England regions is shown both with and without storage scheduling for a second wave of
PPE demand that peaks in mid November. As shown, there is a substantial improvement
in cost savings between the different storage capacities, especially when it is amplified
by a factor of five and 10, resulting in an improvement in cost saving of 15.9% and
7.7% respectively. This means that the challenge of securing PPE during a second wave
can be eased by a considerable amount when additional storage space is provided for
September 28, 2020 15/21stockpiling PPE. The figure also shows that further storage enhancement beyond this
would result in less cost-saving improvements and would therefore have little impact
on the challenge of securing sufficient PPE. This is especially visible when storage is
expanded from 15 times to 20 times the standard storage space. Indeed, enhancing the
PPE storage capacity by 15 times, perhaps by using temporary storage facilities as an
interim solution, results in a relatively steady demand profile as shown in Fig 7.
Fig 7. Effect of increasing storage capacity on the outcome of the game in terms of
aggregate PPE demand. These results are for a second wave where PPE demand peaks
in mid November.
5 Conclusion
In this paper, we developed a game-theoretic model for scheduling PPE supply for
healthcare facilities. In this discrete time dynamic game, healthcare facilities make
stockpiling decisions that minimise their PPE ordering cost. Our model adopts a
centralised-decentralised approach to the PPE supply chain, where a central entity
controls the PPE pricing formula, yet is committed to fulfilling the orders independently
placed by healthcare providers. Based on publicly available COVID-19 hospitalisation
data for NHS England regions, we performed simulations to investigate the impact of
three key factors on PPE security of supply. These factors comprise: (i) the stockpiling
start date, (ii) the time of the peak of a putative second wave, and (iii) the amount
of storage available for medical PPE. The two most critical parameters were found to
be the storage capacity and the stockpiling start date, while the timing of the second
wave has only had a moderate impact on the challenge of securing PPE. Within our
model we observe that enhancing PPE storage capacities by a factor of 15 is sufficient
to considerably lower the peak demand at any given day and effectively minimises1 the
strain on the health care system.
While the shortage of PPE supplies for care home workers during the first peak
in the UK was a topic of great concern, since those shortages were blamed for care
home outbreaks which led to a large number of lost lives [52], we explicitly excluded
1 Please note that this does not imply that dealing with the pandemic in this scenario is rendered
trivial. It rather represents the most favourable way to act during the crisis.
September 28, 2020 16/21this from our study due to the lack of publicly available data. Access to such data
would provide further insights into demand patterns which can potentially influence
the game. Extensions to our work can include introducing supply constraints where
demand cannot always be fulfilled, perhaps by adding a shortage penalty to the cost
function. Also, our model assumes that healthcare facilities can predict their demand
without forecasting errors. This can be enhanced by adding stochastic elements that
capture the uncertainty in demand [28]. The model can also be extended by investigating
scenarios where hospitals can share their resources by having mutual agreements or
using the selfish sharing approach proposed in [53]. This could be via using a scheme
to incentivise matching scheduled regional demand with the production of the supplier.
Another possible extension of this research is to propose a model that finds the optimal
storage capacity for each game player.
Although the above refinements may give greater insight, our model already has
the potential to be a useful tool that can help governments and policy makers in
decision-making in relation to critical PPE supplies.
Acknowledgments
The authors dedicate this work to healthcare workers that have sadly lost their lives
during this pandemic, and would like to thank all of those working on the front-lines,
whether it is in hospital wards, care homes or research laboratories.
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